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LINEAR ELASTIC FRACTURE MECHANICS AS A LIMIT - CASE OF STRAIN - SOFTENING INSTABILITY Alberto Carpinteri Estratto da: MECCANICA Journal of the Italian Association of Theoretical and Applied Mechanics AIMETA VoI. 23 • No.3. September 1988 PITAGORA EDITRICE BOLOGNA

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LINEAR ELASTIC FRACTURE MECHANICSAS A LIMIT - CASEOF STRAIN - SOFTENING INSTABILITYAlberto Carpinteri

Estratto da:

MECCANICAJournal of the Italian Association of Theoretical and Applied Mechanics AIMETA

VoI. 23 • No.3. September 1988

PITAGORA EDITRICE BOLOGNA

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MECCANICA23 (1988), 160-165

LINEAR ELASTIC FRACTURE MECHANICSAS A LIMIT CASEOF STRAIN SOFTENING INSTABILITY*Alberto Carpinteri**

SOMMARIO. Si dimostra come la scala dimensionale, lasnellezza e il grado di iperstaticità, influenzino in modo fon-damentale il comportamento globale di una struttura. Talecomportamento potrà variare da duttile a fragile, allorchési prendano in considerazione l'incrudimento negativoe la localizzazione della deformazione. Il comportamentofragile consiste in una instabilità interna lungo la curvacarico-spostamento, acquistando il ramo discendente unapendenza positiva (snap-back). Tale ramo virtuale puòvenire rilevato sperimentalmente solo nel caso in cui ilprocesso di carico venga pilotato da una funzione monotonacrescente del tempo (es., la larghezza della fessura). Altri-menti, la capacità di carico presenterà una discontinuitàcon un salto negativo. Qualora il comportamento oltre ilcarico massimo venga tenuto sotto controllo sino alla separa-zione completa del provino, l'area compresa tra la curvacarico-spostamento e l'asse dello spostamento, rappre-senta l'energia dissipata nel processo di fessurazione.Viene infine fornita una giustificazione sia alla diminu-zione della resistenza a trazione che all'aumento della tena-cità alla frattura, che si verificano all'aumentare della di-mensione del provino, sulla base dei concetti dell'AnalisiDimensionale. A causa delle diverse dimensioni fisichedi resistenza [FL-21 e tenacità [FL-11, i valori reali ditali due proprietà intrinseche del materiale possono esseredeterminati esattamente soltanto con provini di dimensionerelativamente grande.

SUMMAR Y. Size-scale, slenderness and degree of redundan-cy are shown to have a fundamental impact on the globalstructural behaviour, which can range from ductile to brittlewhen strain softening and localization are taken into account.The brittle behaviour involves an internai instability inthe load-deflectton path, which shows a positive slope inthe softening branch. This virtual branch is revealed onlyif the loading process is controlled by a monotonicallyincreasing time function (e.g., the crack opening=displace-ment). Otherwise, the loading capacity shows discontinuitywith a negative jump, When the post-peak behaviour iskept under control up to the complete structure separa-Non, the area bounded by the load-deflection curve andthe deflection axis represents the energy dissipated in the

* Paper presented at the 8th Nationa1 Conference of the ltalianAssociation of Theoretical and Applied Mechanics, Torino, September29 - October 3,1986.

** Professor of Structural Mechanics, Politecnico di Torino, Torino,Italy.

160

fracture processo A generai explanation of the well-knowndecrease in apparent strength and increase in fracture tough-ness by increasing the specimen size is given in terms ofdimensional analysis. Due to the different physical dimen-sions of strength [FL-21 and toughness [FL-ll, the actualvalues of these two intrinsic material properties may befound exactly only with comparatively large specimens.

1. INTRODUCTION

The curvature localtzation for bent beams is relatedto the fracture toughness of the material. Even if no initialcracks are assumed to exist in the structure, the fracturemechanics concepts are applicable , Size-scale, slendernessand degree of redundancy are shown to exert a fundamentalinfluence on the global structure behaviour, which canrange from ductile to brittle when softening is taken intoaccount.

When a bifurcation (or internaI instability) of the load-deflection equilibrium path occurs, the softening slopebecomes positive. If the loading process is deflection-con-trolled, the loading capacity of the beam shows discon-tinuity with a negative jump. This dreadful event is favouredby large size-scale, high slenderness and/or low degree ofredundancy [Il.

Strain localization and softening are then consideredin order to analyze the three-point bending of rectangularslabs. Even in this case, for large sizes and high slendernesses,the slope of the global softening branch appears tobe posi-tive. This branch is not virtual only if the loading processis controlled by a monotonically increasing time function.When the post-peak behaviour is kept under control up tothe complete structure separation, the area outlined by theload-deflection curve and the deflection axis representsthe product of fracture toughness (~le) by the initialcross-section area.

An explanation of the well-known decrease in apparentstrength by increasing specimen size is given, without makingthe usual statistical Weibull assumptions and assumingthe existence of initial cracks and defects. It has been provedthat the actual strength of the material can be obtained onlywith specimens of infinite size, where the influence of hetero-geneity becomes unirnportant. With the real specimens, anapparent strength higher than the actual one is consistentlyfound.

A cohesive crack model is eventually applied in orderto analyze the three-point bending of initially crackedrectangular slabs. Once more, an internally unstable post-peak behaviour occurs for large sizes. This may be kept

MECCANICA

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under control by the crack mouth opening displacement,which is an increasing function of time, even when load anddeflection decrease following the softening branch withpositive slope. The area marked out by the load-deflectioncurve and the deflection axis is the product of fracturetoughness (<;§ IC) by initialligament area.

An explanation of therecurrent experimental increasein fracture toughness by increasing specimen size is provided,without resorting to the usual non-linear theories and theenergy dissipation in the crack tip plastic zones. It has beenproved that the actual fracture toughness ( KIC) of the rnate-riai can be obtained only with specimens of infinite size.With the real specimens, a fictitious fracture toughness lowerthan the actual one is consistently measured [2].

2. STRAIN-SOFTENING INSTABILITY OF BENT BEAMS

Let us consider a linear elastic moment-curvature relation(Fig. l-a) associated with a linear softening mornent-rotationlaw (Fig. l-b). The area below the moment-rotation curverepresents the energy dissipated by the softening hinge untilthis becomes a free-rotation hinge without resisting momentoThis area is equal to the product of the separation energy0.( strain energy release rate, <;§l C' by the cross-sectionarea, A:.

For the statically determined beam in Fig. 2-a the centraideflection is:

(a) M (b)M

Mu -------

~EI1

Fig. 1. Moment-curvature (a) and moment-rotation (b) Iaw.

~.A .9./3 9./3 9./3

I• t I.----..~__I

(a)

(b)

Fig. 2. Statically determined beam (a) and collapse mechanism (b).

23 (1988)

23 pQ3li=-

648 EI'(2)

E being the Young's modulus of the materia l and I theinertial moment of the cross-section of the beam. If thecross-section is assumed to be rectangular with depth b andthickness t, the ultimate strength load Pu is given by:

tb2 QM =a - =P - (3)u u 6 u3'

where au is the ultimate tensile strength of the material.From eqs. (2) and (3) it is possible to obtain the deflectionin the ultimate strength condition:

23 EuQ2li =--u 108 b '

(4)

(1)

where €u = aulE.If, in the ultimate strength condition, two softening hinges

form at the load application points, the bending momentsin the hinges and, generally , in the beam decrease. At thisintermedia te stage, the hinge rotations increase, whereas thebeam curvatures decrease. Centra! deflection increases ordecreases with the prevalence of hinge rotations or beamcurvatures respectively , Eventually , when hinge rotationsachieve the limit value 'Po' the beam is completely unloadedwithP = O and lì = lio = 'PoQ/3 (Fig. 2-b).

Rearranging of eq. (2) gives:

648 EIP = - - li for li .;;; liu '

23 Q3 '(5)

while the condition of complete load relaxation reads:

(6), When lio > liu' the softening process is stable only if

deflection-controlled , since the slope dP/dli is negative. (Fig. 3-a). When lio = liu' the slope dPldli is not definedand a drop in the loading capacity occurs, even if the loadingis deflection-controlled (Fig. 3-b). Eventually, when lio << lìu' the slope anas becomes positive (Fig. 3-c) and thesame negative jump occurs as that shown in Fig. 3-b. There-fore internai instability occurs for [3]:

E Q2u

b ''PoQ 23--.;;;

3 108(7)

i.e., for:

'Po 23-.;;;-Eu À 36'

where À = Q/b is the beam slenderness. Recalling eqs. (1)and (3), the following is obtained:

(8)

(9)

where:

(lO)

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p

Pu(a)

O ..°u 00 o

p

Pu (b)

O o

p

Pu (c)

li

Fig. 3. Load-deflectìon diagram: (a) extemally unstable; (b) and (c)intemally unstable.

is the brittleness number defined in [4], which is a functionof the material properties and structural size-scale.

Eqs. (8) and (9) provide the brittleness condition:

sE 23 1B=-~-=--.

€u À 432 18.78(11 )

For the statically undetermined beam in Fig. 4, the condi-tion of internal instability was solved by Maier in [1]. Itreads:

KQa = -- ;;;'1.8,

2EI(12)

with K = Muf<po (Fig. l-b). Recalling eqs. (1) and (3), eq.(12) becomes:

(13)

and a brittleness condition is obtained also in this case:

SEB=--~ ---

€uÀ 12x 1.8 21.60

Comparing eqs. (11) and (14), it can be conc1uded that

(14)

~ip ip

~Q/3 Q/3

--\ .•Q/3--II.• •.I••

Fig.4. Statically undetermined beam.

162

the statically determined beam in Fig. 2 is comparativelybrittler than the statically undetermined beam in Fig. 4.Size-scale, slenderness and degree of redundancy are shownto have a fundamental influence on the global structurebehaviour. Brittleness is favoured by large size, high slender-ness and/or low degree of redundancy.

3. STRAIN LOCALIZATION AND APPARENTSTRENGTH

OF INITIALL Y UNCRACKED SLABS

Strain localization is considered in a tliree-point bendingslab of elastic-softening material (Fig. 5, ao = O). A displa-cement discontinuity forms locally when the principalstress achieves the ultimate tensile strength au' At the sametime, cohesive forces rise between the two opposite cracksurfaces. These forces depend on the distance of the interac-ting surfaces: by increasing the distance w, the cohesiveforces decrease until they vanish for w ;;;. wc' In line withthe linear elastic a - € law (Fig. 6-a), a linear a - w .cohesivelaw is assumed (Fig. 6-b). The energy necessary to producea unit crack surface is then:

lwc 1'§/C= o adw="2auwc'

The real crack tip is defined as the point where displace-ment discontinuity is equal to the intetaction thresholdwc' whereas the fictitious crack tip is defined as the pointwhere normal stress attains its maximum value "« andthe displacement discontinuity vanishes (Fig. 7).

The displacement discontinuities on the midline may beex-pressed as follows:

p

t=bl' •I

Fig. 5. Three-poìnt bending slab with initial crack.

(fu °u---------I

b I bI:i I l5w Io: I o:l- l-l/) :2Je I l/)

I1 I

Wcl'u ooSTRAIN. < OPENING. W

(.) (b)

Fig. 6. Stress-strain (a) and stress-crack opening displacement (b) consti-tutive laws.

MECCANICA

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t.a

!I

)

Fig. 7. Stress distribution across the cohesive zone: real and fictitiouscrack tips.

(a

J

w(x) ~ tK(x, y) u(y) dy + C(x)P + P(x),

for O.;;;x < b,(16)

where K and Care the influence functions of cohesiveforces and extemal load respectively, and rare the crackopenings due .to the specimen weight. If a stress-free crackof length a has developed with a cohesive zone of lengthèa, thé following additional conditions are to be takeninto account:

a(y) = O, for O';;;y «;«,

a(y)=au [1- w~)], fora';;;y';;;(a+t.a),

w(x) =0, for (a+~a)';;;x<b.

Eqs, (16) and (17) can be rearranged as follows:.:w(x) = Kix, y)

a[

W(y)]1- -;- au dy +

c

+ [ K(x,y)a(y)dy+C(x)P+r(x),a+c,a

(17-a)

(17-b)

(17 -c)

for O.;;; x.;;; (a + tsa), (l8~a)

(18-b)w(x) = O, for (a + ~a) .;;;x <b.

Eqs. (18-a) and (l8-b) are two integral equations for theunknown functions a and w. These equations can be discreti-zed by considering n points on the midline between thebottom and the top of the beam [5 l.

The additional unknown parameter P can be determined,since both equations are valid for x = a + èa, i.e. at thefictitious crack tipo

On the other hand, the beam deflection is given by:

0= fb C(y) a(y) dy+ DpP +o,o

(19)

where D is the deflection for P = l and D is the deflectionp 'fdue to the specimen weigth.

A numerical procedure is implemented to simulate aloading process whereby the fictitious crack depth is increas-ed step by step [5]. Real (or stress-free) crack depth,extemalload and deflection are obtained at each step as a result ofan iterative computation. Some dimensionless load-deflectiondiagrams for a concrete-like material are plotted in Fig. 8-a,with ao/b = 0.0, €u = 0.87 X 10-4, Il = 0.1, t = b, Q = 4b,and by varying the non-dirnensional number [4]:

t'§ICS =~-E a b

u 2b(20)

The specimenbehaviour is brittle (or internally unstable)for low sE numbers, i.e., for 10w fracture toughnesses t'§IC'

high tensile strengths, a , and/or large sizes b. For sE <u _

~ 10.45 X 10-5, the P-O curve exhibits a positive slope inthe softening branch and a catastrophic event occurs if theloading process ìs deflection-controlled, This indentingbranch is not virtual only if the loading process is controlledby a monotonically increasing time function , such as thedisplacement discontinuity across the crack. When thepost-peak behaviour is kept under control up to the completestructure separation, the area bounded by the Ioad-deflectìon

N 0.36.c,~Ò«o..J

~ 0.24zov;~::;;c 0.12

0.48r---------------~71 0.16R (a)

ap

o

0.12

0.08

0.04

1~ ~4 ~6OIMENSIONLESS DEFLECTION. 6/b. 16'

'o/b; 0.0

A SE c 2.09 E - 58 'E ; 4. 18E - 5C sE ; 6.27 E ~ 5D SE = 8.36E - 5E SE = 10.4SE - 5F SE ; 20.90E - 5G lE = 41.80E-SH sE ; 62.70E - 5I lE = 83.S9E - 5L sE = 1.04E-3M sE 2.08E - 3N SE 4.18E-3O SE 6.27E - 3P sE 8.36E - 3a sE = 10.45E - JR sE = 20.90E - J

(b)

M

6 12 18DIMENSIONLESS DEFLECTION. o/b. 103

24

23 (1988)

Fig. 8. Load-deflection diagrarns of initiaIly uncraked slabs: (a) Q = 4b; (b) Q = 16b.

163

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3

.!!..

~2.~

'i,.1,L-===========-'o/b-O.O

DIMENSIONLESSSIZE. ba/ !f,C 11031

Fig. 9. Decrease in apparent strength with increasing specimen size.

curve and the deflection-axis is the product of fracturetoughness, é§le' by initial cross-section area, bt ..

The diagrams in Fig. 8-b refer to an higher beam slen-derness, Q = 16 b. The same brittleness increase with decreas-ing sE is obtained as previously, but in this case it is easierto achieve the internal instability of the system, when sE ::;

< 62.70 X 10-5•

- The maximum loading capacity p<~~ of initially un-cracked specimens with Q == 4b is obtained frorn Fig. 8-a.On the other hand, the maximum load P<~~x of ultimatestrength is given by:

2 a' tb2uP(3)

max(21)

3 Q

The values of the ratio p( 1) 1.[13) may also be regardedmax maxas the ratio of the apparent tensile strength, al (given bythe maximum load p(1) and applying eq. (21» to themaxactual tensile strength au (considered to be a material con-stant). It is evident from Fig. 9 that the results of the cohe-sive crack model tend to be similar to those of the ultimate.strength analysis for low sE values:

lim p(1) ==p(3).'E"" o max max

(22)

0.12 ...------------------,

0.09"=,~o'«s~w...JZoVizw:;C

0.06

0.03

Therefore, only for comparatively large specimen sizescan thetensile strength au be obtained as au == ar With theusual laboratory specimens, an apparent strength higherthan the actual one is always found .

4. COHESNE CRACK PROPAGATION AND FICTITIOUSFRACTURE TOUGHNESS OF INITIALLY CRACKEDSLABS

Themechanical behaviour of three-point bending slabswith initial craks (Fig. 5) is .investigated on the basis ofthe same cohesive numerical model described in the previoussection. Some dimensionless load-deflection diagrams aregiven in Fig. lO, for ao/b == 0.5,è u == 0.87 X 10-4, V ==

== 0.1, t == b, Q == 4b, and for various values of the brittle-ness number sE (see eq. (20». The initial crack makes thespecimen bahaviour more ductile than in the case of initiallyuncracked specimen. Far the set of sE numbers consideredin Fig. lO, no internal instability occurs.

The area delimited by the load-deflection curve and thedeflection-axis is the product of fracture toughness, é§ le'by initial ligament area, (b - ao )t. The areas below theP-O curves are thus proportional to the respective SE numbers,as is shown in Fig. lO as well as in Fig. 8. This simple resultis based on the assumption that energy dissipation occursonly on the fracture surface, whereas in reality energyis also dissipated in a damage volume around the crack tip,as was assumed by Carpinteri and Sih in [6~On the otherhand, the cohesive crack assumption is more than acceptablefor slender beams, where bending prevails over shear andenergy dissipation is concentrated in a very narrow crackband [7].

The maximum loading capacity p(1) according tothemaxcohesive crack model, is obtained frorn Fig. lO. On theother hand, the maximum load p(2) of brittle fracturemaxcan be obtained from the Linear Elastic Fracture Mechanics

G

.o/b = 0.5

A se = 2.09E - 5B se = 4.18E - 5C se = 6.27E-5D se = 8.36E - 5E se = 10.45E - 5I: se = 20.90E - 5G se = 41.BOE-5H se = 62.70E - 5I se = 83.59E - 5L se = 1.04E - 3M se = 2.08E - 3N se = 4.1BE_3O se = 6.27E - 3P se = 8.36E - 3Q se = 10.45E - 3R se = 20.90E - 3

0.6 1.20.00 L--_"'--_-'-_-'-_-'--_-'-_---'-_---I_---'

0.0 2.4

DIMENSIONLESS DEFLECTION. 6/b. 103

1.8

Fig. lO. Load-deflection diagrams of initially cracked slabs (Q = 4h, aO = h/2).

164 MECCANICA

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20

~ _-_ .•..........•.•.....•.•. _-_ __ .._--------

~f:-;:::-------

ao'b =O.1 ._-----'o/b=0.3 --ao/b=0.5 --

100 %-~~E!!:

60::~..60

40

o L- ~ ~ ~----~----~~~O 2 3 4 5 l/SE

DIMENSIONLESS SIZE. bo/ ~'C (103)

Fig. Il. Increase in fictitious fracture toughness by increasing thespecimen size.

equation [8]:

K = P~jxQ f (abo),

IC tb3/2

f(~) = 2.9 (a;t - 4.6(a~r fa )5/2+ 21.8 \ ;

_ 37.6 {a;t2 + 38.7 ~a;r2with KIC = V <§ICE (plane stress condition).

The values of the ratio p< 1) /p(2) are provided in Fig. Ilmax maxas functions of the inverse of the brittleness number "s:This ratio can also be regarded as the ratio of the fictitiousfracture toughness (given by the maximum load p(~~x )

to the actual fracture toughness (considered to be a material

(23)

REFERENCES[I] MAIER G., Sul comportamento flessionale instabile nelle travi

inflesse elastoplastiche, Istituto Lombardo, Accademia di Scienzee Lettere, Rendiconti, Classe di Scienze (A), VoI. 102, pp. 648-677,1968.

[2] CARPINTERI A., No teh sensitivity in fracture testing of aggregativematerials, Engineering Fracture Mechanics, 16, pp. 467481,1982.

[3] CARPlNTERI A., Limit analysis for elastic-softening structures:scale and slenderness influence on the global brittleness, Structureand Crack Propagation in Brittle Matrix Composite Materials,Euromech Colloquìum No. 204, November 12-15, 1985, Jablonna(Warsaw),edited by A.M. Brandt, Elsevier Applied Science Publì-shers, pp. 497-508,1986.

[4] CARPINTERI A., lnterpretation of the Griffith instability as abifurcation of the global equilibrium, Application of Fracture

. 23 (1988)

constant).It is evident that, for low SE numbers, the results of the

cohesive crack model are similar to those of Linear ElasticFracture Mechanics:

(24)

and, therefore, maximum loading capacity can be predictedby applying the simple condition KI = KIC' It appears thatthe actual fracture toughness KIC of the material can beobtained only with very large specimens. In fact, with lab-oratory specimens, a fictitious fracture toughness lowerthan the actual one is always measured.

5. CONCLUSIONS

Recalling Figs. 9 and Il once again, it can be statedthat the smaller the brittleness number sE' the more accura-tely the bifurcation reproduces the perfectly brittle ultimatestrength instability (ao = O) or the Linear Elastic FractureMechanics instability (ao '* O).

Therefore, ultimate tensile strength au or fracture tough-ness KIC can be obtained exactly only with very large (ini-tially uncracked or cracked respectively) specimens. On theother hand, the criticaI value <§ IC of strain energy releaserate may be derived from the area delimited by the load-deflection curve and the deflection-axis for any specimensize.

ACKNOWLEDGEMENTS

The numerical results reported in the present paper wereobtained in a joint research program between ENEL-CRIS-Milano and the University of Bologna.

Received: November 14, 1987; in revised form: Mareh 14, 1988.

Mechanics to Cementitious Composites, NATO Advanced ResearchWorkshop, September 4-7, 1984, Northwestern University, editedby S.P. Shah, Martinus Nijhoff Publishers, pp. 287-316, 1985.

[5] CARPlNTERI A., DI TOMMASO A., FANELLI M.,lnf1uence ofmaterial parameters and geometry on cohesive eraek propagation;International Conference on Fracture Mechanics of Concrete,Lausanne,October 1-3, 1985, edited by F.H. Wittmann, ElsevierScience Publishers B.V., pp. 117-135,1986.

[6] CARPlNTERI A., SIH G.C.,Damageaecumulation and crack growtnin bilinear materials with softening, Theoretical and Applied Frac-ture Mechanics, l, pp. 145-159,1984.

[7] BAZANT Z.P., CEDOLIN L., Blunt erack band propagation in finiteelement analysis, Journal of the Engineering Mechanics Division,ASCE, 105,pp. 297-315,1979.

[8] Standard Method of Test for PIane Strain Fracture Toughnessof Metallic Materials, C 399-74, ASTM.

165